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SUMMARY:An AI-Assisted Proof in Spectral Geometry - Jonas Henkel (Philipps
 -Universität Marburg)
DTSTART:20260420T141500Z
DTEND:20260420T143000Z
UID:TALK246202@talks.cam.ac.uk
DESCRIPTION:Computing the spectrum of the Hodge-Laplacian on homogeneous s
 paces is a challenging problem. In a joint project with E.\,A. Lauret\, we
  studied the spectrum on $1$-forms for general left-invariant metrics on $
 S^3 cong SU(2)$ and its quotient $SO(3)$. We successfully computed the exp
 licit full spectrum for Berger spheres and formulated a highly plausible c
 onjecture for the first eigenvalue of general homogeneous metrics. However
 \, despite extensive numerical evidence\, a rigorous abstract proof for th
 is general case remained difficult to find for over a year. &nbsp\; &nbsp\
 ; &nbsp\; &nbsp\; &nbsp\;\nJust before submitting our manuscript\, we deci
 ded to run a final test using the newly released AI model ChatGPT 5.4 Pro 
 (a premium model costing $200 per month\, limited to 15 prompts). Provided
  with a single prompt\, the model reasoned completely on its own for 100 m
 inutes. Surprisingly\, it came up with an elegant and flawless abstract pr
 oof for our conjecture. Crucially\, the AI took a completely different pat
 h from our previous manual attempts: by defining a clever diagonal transfo
 rmation to pass to the round metric and using the self-adjoint Curl operat
 or\, it successfully established the required lower bound. Our research gr
 oup reviewed the generated proof over the following days and confirmed its
  validity. &nbsp\; &nbsp\; &nbsp\; &nbsp\; &nbsp\;\nThis proposed flash ta
 lk will share a brief\, anecdotal report of this remarkable AI-assisted di
 scovery. It will show the mathematical setup of the problem and demonstrat
 e how advanced LLMs can navigate intricate algebraic and geometric barrier
 s\, offering a glimpse into the potential of AI as a reasoning assistant i
 n pure mathematics.
LOCATION:Seminar Room 1\, Newton Institute
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